## Stability of parabolic Harnack inequalities on metric measure spaces.(English)Zbl 1102.60064

The paper extends the results of M. T. Barlow and R. F. Bass [Trans. Am. Math. Soc. 356, No. 4, 1501–1533 (2004; Zbl 1034.60070)] and M. T. Barlow [in: Surveys in Differential Geometry 9, 1–25 (2004; Zbl 1070.53039)], to the case of a metric measure space with a local regular Dirichlet form. A necessary and sufficient condition for a parabolic Harnack inequality (with global space-time scaling exponent $$\geq 2$$) to hold is proved. This parabolic Harnack inequality is stable under rough isometries. It follows that such a Harnack inequality holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

### MSC:

 60J35 Transition functions, generators and resolvents 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31C25 Dirichlet forms 60E15 Inequalities; stochastic orderings

### Citations:

Zbl 1034.60070; Zbl 1070.53039
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